Catastrophic generator matrix Given the generator matrix $$G(z) = [z^3+1 \ \ \ \ \ \ \ z^3+z^2+z+1]$$ Why is G(z) catastrophic? I know that a generator matrix for a rate k/n convolutional code is called catastrophic, if there exists an information sequence $m(z) \in F_2^k[[z]]$ with infinitely many nonzero digits, that results in a codeword $c(z) \in F_2^n[[z]]$ with only finitely many nonzero digits. Thanks for any help!

A polynomial generator matrix is catastrophic if and only if its (full rank) minors have a non-trivial common divisor. Here the entries have a common divisor $$ \gcd(z^3+1,z^3+z^2+z+1)=\gcd((z+1)(z^2+z+1),(z+1)^3)=z+1, $$ so this is, indeed, the case.

Because this is a $(2,1)$-code, we immediately see that the infinite weight input $$ m(z)=\frac1{1+z}=1+z+z^2+z^3+\cdots $$ will serve as an example. For a higher rank code there would be a bit more work to do.

Leaving it to you to find the corresponding finite weight codeword.